1. Introduction
A topological index is a numeric value that represents the symmetry of a molecular structure. Indeed, it is a mathematical classification of a chemical graph that offers a mathematical function in a quantitative structure–property relationship (QSPR). It links numerous physicochemical characteristics of molecular structured chemical substances, such as the strain energy, stability, and boiling point. Numerous characteristics of chemical compounds having a molecular structure can be examined using several kinds of topological indices. In
H. Wiener presented the notion of the first topological index in researching the boiling point of paraffin, which he called the path number [
1]. As a result, it was dubbed the Wiener index, and it was the moment that the idea of topological indices began.
Polya’s [
2] concept of computing the polynomials was used by many chemists to identify the molecular orbitals of the unsaturated hydrocarbons. In this context, the spectrum of a graph has been extensively examined. According to [
3], Hosoya used this idea in 1988 to determine the polynomials of several chemical structures, which were labeled the Hosoya polynomials and attracted widespread attention. In 1996, Sagan et al. [
4] then retitled the Hosoya polynomial as the Wiener polynomial, although most experts still refer to it as the Hosoya polynomial. The information regarding distance-based graph invariants may be obtained from the Hosoya polynomial. In [
5], Cash noticed a connection between the hyper Wiener index and the Hosoya polynomial. Estrada et al. [
6] investigated several interesting applications of the expanded Wiener indices.
The graphs presented in this paper are all simple graphs, meaning they have no loops nor multiple edges. Suppose
is a finite group. The power graph represented by
of
is a graph, in which
is its node set and two unlike elements are edge connected if and only if one of them is an integer power of the other. In [
7], Kelarev and Quinn discussed the approach of directed power graphs related to groups and semigroups. Later, the authors of [
8] illustrated the power graph
of a semigroup
and identified the class of semigroups, whose power graphs are the complete graphs. Furthermore, they discussed that for any finite group
, the associated power graph is the complete graph if and only if the group
is cyclic of order one or
, where
p is any prime and
In the current literature of theory of graphs, the power graph is now an exciting topic in several branches of mathematics, that is group theory, ring theory, and Lie algebra. Cameron et al. [
9] discussed the matching numbers and gave the upper, as well as the lower bounds for the perfect matching of power graphs of certain finite groups. They also derived a formula of matching numbers for any finite nilpotent groups. The authors of [
10,
11,
12,
13] presented an overview of finite groups with enhanced power graphs that enable the formation of a perfect code. They further established all possible perfect codes of the proper reduced power graphs and gave a necessary and sufficient condition for graphs having perfect codes. In [
14], the authors concentrated on the power indices graph and classified all such graphs in some specified categories.
The authors of [
15] examined the maximum clique and found the largest number of edges of power graphs for all the finite cyclic groups. Sriparna et al. [
16] deliberated about the node connectivity of
, whenever
n is the product of some prime numbers. Furthermore, several other researchers inquired about different concepts of algebraic graphs; for instance, see [
17,
18,
19,
20] and the references therein.
A matching or an independent edge set is the collection of edges that share no nodes. When a node is coincident with one of the matching edges, it is referred to as matched. Otherwise, there is an unmatched node. The maximum number of matchings in a graph is referred to as the
Z-index or Hosoya index. Hosoya [
21] first proposed the
Z-index in 1971 and then extended the topological index as a common tool for quantum chemistry in [
22]. It has since been shown to be useful in a variety of molecular chemistry problems, including the heat of vaporization, entropy, and the boiling point. The
Z-index is a well-known topological index example that has significant relevance in combinatorial chemistry. Considering numerous graph structures, many researchers investigated extremal problems in regard to the
Z-index. Extremal characteristics of different graphs, unicyclic graphs, and trees were extensively studied in [
23,
24,
25,
26].
In this article, we represent the cyclic group of order n, the generalized quaternion group , and the dihedral group of order and , respectively. It is very challenging to calculate the (reciprocal) Hosoya polynomial, as well as the Z-index of power graph of a group . In this regard, we provide both the Hosoya and the reciprocal Hosoya polynomials and also discuss the Z-index of the power graph of a group , when is or .
There are still several gaps in the current study about the determination of the Hosoya polynomials, the reciprocal Hosoya polynomials, and also the Z-index or Hosoya index of the power graphs of a finite cyclic group , the dihedral group , and the generalized quaternion group . We look at one of these problems in this article.
2. Basic Notions and Notations
This part reviews several fundamental graph-theoretic properties and well-known findings that will be important later in the article.
Suppose is a simple finite undirected graph. The node and edge sets of are represented by and respectively. The distance from to in a connected graph denoted by is defined as the shortest distance between and . The total number of nodes, denoted by , is said to be the order of . Two nodes and are adjacent if there is an edge between them, and we denote them by otherwise The valency or degree represented by of a node is the collection of nodes in , which are adjacent to . A path having length is known as a geodesic. The largest distance between a node and any other node of is known as the eccentricity and is denoted by . The diameter denoted by of is the largest eccentricity among all the nodes of the graph Furthermore, the radius symbolized by of is the lowest eccentricity among all the nodes of the graph
Suppose
is a graph of order
n. According to Hosoya, the polynomial of
with a variable
y is defined as follows:
The coefficient
represents the number of pairs of nodes
so that
where
. Ramane and Talwar [
27] proposed the reciprocal status Hosoya polynomial of
, which is given as:
where
is referred to as the transmission or the reciprocal status of a node
w.
Suppose
and
are two connected graphs, then
is the join of
and
whose node and edge sets are
and
, respectively. A complete graph is a graph that has an edge between any single node in the graph, and it is symbolized by
. Other unexplained terminologies and notations were taken from [
28].
Definition 1. Assume that is a group. Then, the center of is given as: The dihedral group
is the group of symmetries, and its order is
, where
. The presentation of a dihedral group is given by:
Throughout this paper, we mean
where
and
p is any odd prime number. We now split
as follows:
where
and
. Since
and
, for all
, where the identity
e is connected to every other node in its power graph, the subgraph induced by
is a complete graph
.
Furthermore, the presentation of generalized quaternion group
of order
for
, where
, is given as:
We now split
as follows:
and
. Since
is cyclic, its induced subgraph is complete, and it is denoted by
. A remarkable feature of
is that the involution
and the identity
e are adjacent to every other node in their power graph. Several researchers [
29,
30,
31,
32] have analyzed the complete description of the above-mentioned groups and their power graphs. The first survey paper on power graphs was published in 2013 [
33], and the most recent study was [
34].
Proposition 1 ([
31])
. The power graph of satisfies: Proposition 2 ([
31])
. The power graph of satisfies:where represents the n copies of . 5. Hosoya Index
The Hosoya index of the power graphs of finite groups is examined in this section. On a graph with
n nodes, the greatest feasible value of the Hosoya index is provided by the complete graph
[
35]. In general, the Hosoya index of
where
, is as follows:
this may be seen concerning the entire non-void matchings specified in
Table 3, whereas
represents the cardinality
i matchings, where
Table 3.
The total non-void matchings in .
Table 3.
The total non-void matchings in .
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Proof of Theorem 3. From the structure of , the identity e is the only node that is connected to every other node in . Therefore, there are three types of edges in , i.e.,:
- Type 1:
, for
- Type 2:
, for
- Type 3:
, for
Since we know that the subgraph induced by is complete, i.e., , thus, has two distinct types of matchings:
- T1:
For each type, the number of matchings may be calculated as follows: Due to the fact that the edges of Type 1 and Type 2 are the edges of a complete graph
, which is induced by the nodes in
, so the number of matchings in this type can be obtained by counting the matchings in
, which are given in
Table 4, where
denotes the number of matchings of order
i, for
- T2:
Every matching of this kind may be generated by substituting one edge of Type 3 for any edge of Type 1. Given that every Type-1 edge is also an edge of
, which is induced by the nodes of
, so every matching of Type-1 edges is also a matching of
. The total number of certain matchings is described in
Table 5, where for any
,
signifies the total number of matchings of order
i.
Given that Type 3 has m edges, the essential matchings may be produced as:
Next, using the sum rule, the entire matchings in each order (matchings
+ matchings
) may be calculated as: The number of matchings of order one is as follows:
Order two has the sequel number of matchings:
Order three has the following number of matchings:
Order four has the following number of matchings:
Generally, order
i has the following number of matchings:
where
Thus, the Hosoya index of
is given by:
□
Table 4.
The total non-void matchings in .
Table 4.
The total non-void matchings in .
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Table 5.
The total non-void matchings in .
Table 5.
The total non-void matchings in .
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Proof of Theorem 4. Using Proposition 2, the node set is , where . Therefore, we have the sequel types of edges in :
- Type 1:
, for any
- Type 2:
, for any
- Type 3:
, for any
- Type 4:
, for any
- Type 5:
, for any where
Seven cases of matchings occur amongst the edges of according to the following categories:
Matchings amongst the Type-1, Type-2, as well as Type-3 edges;
Matchings amongst the Type-4 edges;
Matchings amongst the Type-5 edges;
Matchings amongst the Type-1 and Type-4 edges;
Matchings amongst the Type-3 and Type-4 edges;
Matchings amongst the Type-4 and Type-5 edges;
Matchings amongst the Type-1, Type-2, Type- and Type-5 edges.
The following method computes all the types of matchings as mentioned above:
As we know that the subgraph induced by
is complete, that is,
, so all the Type-1, Type-2, and Type-3 edges are exactly the edges of
, and all the matchings between these edges are counted in
Table 6, where
denotes the number of matchings of order
i, where
;
For , let indicate the number of order i matchings:
- For
: The number of Type-4 edges that are which is equal to the number of order one matchings. Consequently,
- For
: Suppose
is a Type-4 edge with
and
for a fixed
Then, in addition to the edge
e, every edge of Type 4 with one end in
and another in
forms a matching of order two. As a result,
Hence, there is no order larger than two matching in this situation;
Type 5 has n edges, none of which have a similar node. Thus, for any order i, there exists a matching such that . Suppose represents the number of order i matchings. Then,
Assume that
represents the number of order
i matchings, where
Thus, in this context,
. There are no Type-1 edges that connect a node to any Type-4 edge in
. Hence, we may obtain a matching in this situation by joining every matching of Type-1 edges to any matching of the Type-4 edges. The edges of Type 1 are also the edges of
, and there are
matchings of order
ℓ between them, where
Every
can be found in
Table 6. In between the edges of Type 4, there are
and
matchings having one and two orders, respectively.
As a result of the product rule, we obtain:
Furthermore, when
then:
For
,
represents the total matchings of order
i. Then,
. We can only utilize matchings of order one between the edges of Type 4 in this case. Otherwise, we are unable to employ any Type-3 edge, since both kinds of edges often share the nodes in
. As a result, we can only obtain matchings of order two in this case. Assume that
is the order one matching amongst the Type-4 edges with
, for
Then, any Type-3 edge that is non-adjacent with
can lead to construct an order two matchings. Given that there are
such Type-3 edges, every of which may be utilized in every one of
matchings of order one amongst Type-4 edges, so we obtain:
For
,
, denote the number of order
i matchings, then
. To find matchings, both matchings of order one and two between the edges of Type 4 will be considered and any matching of order
ℓ between the edges of Type 5, where
. Thus, by counting these matchings using the product rule, we obtain:
and for
Given that the edges of Type 1, Type 2, as well as Type 3 are also the edges of
, which is induced by
, so we may utilize them to identify matchings between the edges of Type 5 and the edges of
. Let
be the number of order
i matchings. Then,
. Because no edge of Type 5 shares a node with an edge of
, this corresponds to each matching of the edges of Type 5. Therefore, each matching of the edges of
can be used to determine a match in this situation. Since there exist
matchings of the cardinality
among the edges of
, as shown in
Table 6, as well as
matchings of the order
among the Type 5 edges, thus, in this example, the greatest order of a matching is
. As a result, we may determine
, for
, as follows:
As a consequence, by the sum rule, the Hosoya index of
is as follows:
where:
for
,
□
Table 6.
The total non-void matchings in .
Table 6.
The total non-void matchings in .
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